
// Make newform 490.2.l.c in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_490_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_490_l_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_490_2_l_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_490_2_l_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [2401, 0, 3038, 0, 1933, 0, 1438, 0, 852, 0, 266, 0, 61, 0, 10, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1302910, 0, 786719, 0, 866714, 0, 343988, 0, 96110, 0, 20266, 0, 2802, 0, 171, 0], [-204554077, -163267041, -120098303, -4504619, -131660244, -2623012, -52276470, -2242730, -13637974, -1140202, -2335656, -291088, -319853, -94829, -6023, -6279], [-204554077, 163267041, -120098303, 4504619, -131660244, 2623012, -52276470, 2242730, -13637974, 1140202, -2335656, 291088, -319853, 94829, -6023, 6279], [1019822349, 0, 189804581, 0, 516320528, 0, 433352890, 0, 144339958, 0, 35853172, 0, 6218761, 0, 657161, 0], [-1273004222, 0, -843429583, 0, -445540384, 0, -373946420, 0, -125675214, 0, -30938216, 0, -5366258, 0, -623003, 0], [8000085009, 6797535927, 220726331, 1618097653, 128527588, 3441488944, 109893770, 2888475470, 55869898, 995989274, 14263312, 238976156, 4646621, 41450603, 307671, 4174573], [8000085009, -6797535927, 220726331, -1618097653, 128527588, -3441488944, 109893770, -2888475470, 55869898, -995989274, 14263312, -238976156, 4646621, -41450603, 307671, -4174573], [-11101746201, -15782597033, -2889806479, -10077798427, -5620645072, -7454251516, -4717462610, -4444289310, -1650388502, -1397075246, -390296228, -316254784, -67697189, -53207257, -6673919, -5413027], [11101746201, -15782597033, 2889806479, -10077798427, 5620645072, -7454251516, 4717462610, -4444289310, 1650388502, -1397075246, 390296228, -316254784, 67697189, -53207257, 6673919, -5413027], [0, 42231189, 0, 8929484, 0, 21381008, 0, 17945290, 0, 6079906, 0, 1484692, 0, 257521, 0, 26590], [0, -477500443, 0, -303303017, 0, -226254436, 0, -133651610, 0, -41588666, 0, -9597064, 0, -1553547, 0, -154017], [-17501784573, 9719272549, -16785607657, 6928535561, -8860887056, 1535163128, -7437029530, 3102633990, -2508526286, 1167320938, -615297844, 313306372, -106723897, 58163261, -12843957, 7738601], [0, -6839548919, 0, -4832044771, 0, -1033713348, 0, -2161262850, 0, -802913678, 0, -212402552, 0, -39396031, 0, -5179651], [-1821562113, -1179772034, -1214738142, -610473801, -642101046, -355475148, -539065950, -427936390, -183171156, -154522158, -44968654, -39448752, -7955037, -7115276, -939402, -850941]];
    Rf_basisdens := [1, 1, 1020740, 224562800, 224562800, 785969800, 392984900, 11003577200, 11003577200, 11003577200, 11003577200, 50016260, 239208200, 11003577200, 5501788600, 785969800];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_490_l();"
function MakeCharacter_490_l()
    N := 490;
    order := 12;
    char_gens := [197, 101];
    v := [3, 10];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_490_l_Hecke();"
function MakeCharacter_490_l_Hecke(Kf)
    N := 490;
    order := 12;
    char_gens := [197, 101];
    char_values := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 1, -1, 0, -1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 1, 1, 2, 0, -1, 1, -2, 2, 0, 0, -2, 0, -2], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [3, -1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, -1, 2, 0, 0, -3, 1, -1, 2, 1], [-1, 2, -2, 0, 0, 1, -2, 0, 0, 0, 0, -1, 1, -1, 2, 0], [0, -1, 0, 0, 0, 0, 0, -4, 4, 0, 0, 1, 1, 0, 0, 0], [-2, 0, -2, 2, 2, 0, -2, 0, 1, 0, 0, 0, 0, -1, 0, -1], [0, -2, -2, 2, 0, 1, -1, -1, 0, -1, 0, 1, -1, 0, -1, -2], [-1, 0, -2, 0, 0, 2, -2, 2, -2, 0, 0, 0, 0, 4, 0, 4], [0, 1, -1, 0, 1, -1, 0, 0, -1, 0, -1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, -1, 1, 3, 0, 1, 0, -1, 1, 0, 1, -2], [1, -1, 1, 0, -1, -2, 5, 0, 1, 0, -2, 4, -2, 0, -5, 0], [0, 2, 0, 0, 0, 0, 0, -4, 2, 0, 0, 0, 0, 2, 2, -2], [0, 0, 0, 0, 0, 1, -1, 2, 4, 0, 0, 0, 0, -2, 0, -2], [0, 0, 0, 1, 0, 0, -4, -1, 0, -2, 0, 4, 0, 0, -4, 3], [0, 0, 2, 1, 1, 2, 0, 4, 4, 1, -1, 0, 0, 0, 0, 0], [1, -1, 1, 0, -6, 0, 1, 0, -2, 0, 0, 0, 0, 4, -1, 0], [0, 0, 0, 4, -4, 0, 0, -2, -2, -2, -2, 0, 0, 4, 2, -4], [2, 2, -2, 0, 3, 2, 4, 0, 5, 0, 3, 2, 2, -10, -4, 0], [0, 0, 0, 0, 0, 0, 0, -3, 4, -2, -2, -3, -1, -1, 1, 1], [1, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, -1, 0, 0, 2, 8], [-5, 0, 7, -3, -3, 0, 1, 1, -1, 0, 0, 0, 0, 2, 0, 2], [-3, 0, 0, 0, 0, 4, -1, 0, -11, 0, 1, 2, 4, 5, 1, 0], [3, 2, 2, -6, 0, -1, 4, 4, 0, 3, 0, -1, 1, 0, 4, 7], [0, 1, 0, 2, -2, 0, 0, -2, -6, -4, -4, 2, 2, 8, -1, -8], [-2, -3, 3, 0, -4, 3, 0, 0, -4, 0, 4, 2, 3, 0, 0, 0], [-5, -1, -1, 1, 0, -1, 0, 10, 0, 1, 0, -5, 1, 0, 0, 0], [3, 0, 0, 0, 0, -3, 0, 1, 4, -2, 2, 0, 0, -3, 0, -3], [0, 0, 0, -4, 0, 0, -2, 4, 0, 8, 0, 2, 0, 0, -2, 8], [0, -2, 0, -1, 1, 0, 0, -5, -5, 1, 1, 0, -2, 10, 0, -10], [0, 6, 0, -6, 6, 0, 0, 5, 3, 3, 3, -3, 3, -8, 9, 8], [0, 0, 0, 3, 3, 0, 0, -3, -2, -6, 6, 0, 0, -1, 0, -1], [-2, -2, -2, -5, 0, 0, -2, 7, 0, 0, 0, 0, 0, 0, -2, 6], [8, -4, 4, 0, 4, -2, 10, 0, 2, 0, 2, 2, -2, 2, -10, 0], [2, 1, 1, -2, 0, -1, 4, 5, 0, 2, 0, -2, 1, 0, 4, 10], [0, 0, 0, 0, 0, -1, 1, 0, 1, 0, -5, 1, -1, 2, -1, 0], [0, -1, 0, -1, 1, 0, 0, 1, 3, 2, 2, 1, -2, -4, -2, 4], [-1, 0, 3, 3, 3, -3, -2, -2, 2, -3, 3, 0, 0, -4, 0, -4], [-2, 0, -8, 2, 2, 4, -6, 4, 1, -1, 1, 0, 0, 3, 0, 3], [0, 0, 0, 0, 2, 0, -5, 0, 10, 0, 4, -5, 0, -12, 5, 0], [6, -2, -2, 5, 0, -2, 0, 0, 0, 5, 0, 6, 2, 0, 0, 0], [0, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, 6, 0, 0, 6, 0], [8, 0, -3, 0, 0, -3, 0, 5, 5, 0, 0, 0, 0, 0, 0, 0], [2, -2, 2, 0, -2, -2, 4, 0, -4, 0, -2, 2, -2, 8, -4, 0], [1, -5, -5, 8, 0, 0, -2, 0, 0, 0, 0, 3, 0, 0, -2, -4], [0, 0, 0, 0, 0, 0, 0, 2, -6, -2, -2, -4, -4, 4, 0, -4], [9, 0, 0, 0, 4, 0, 9, 0, 2, 0, 2, 0, 0, -6, -9, 0], [0, 0, 0, 2, -2, 0, 0, 1, -1, 2, 2, 14, 0, 0, 0, 0], [13, 0, -1, 2, 2, 0, 6, -8, -3, 0, 0, 0, 0, -5, 0, -5], [2, 0, 3, -5, -5, -3, 4, 4, -4, 5, -5, 0, 0, 8, 0, 8], [1, 0, 0, 0, 0, 3, -4, 4, 0, 6, 0, 5, -3, 0, -4, 2], [0, 0, 0, 0, 3, 0, 3, 0, -2, 0, 6, 3, 0, -1, -3, 0], [0, -7, 0, 2, -2, 0, 0, 6, -4, 0, 0, 8, 0, -2, 1, 2], [8, 0, 0, 0, 0, -2, -6, 0, 4, -4, 4, 0, 0, -4, 0, -4], [-3, -3, -3, 2, 0, 6, 3, -2, 0, -4, 0, -6, -6, 0, 3, -6], [-10, 0, -1, 2, 2, -1, 0, -7, -7, 2, -2, 0, 0, 0, 0, 0], [10, 2, -2, 0, 4, 0, 4, 0, 6, 0, 0, -6, 0, 2, -4, 0], [-1, 7, -7, 0, 5, 7, -2, 0, 2, 0, 5, -1, 7, -4, 2, 0], [0, 3, 3, -7, 0, -3, 0, -1, 0, 7, 0, 0, 3, 0, 0, -2], [2, 0, -2, -1, -1, 0, 0, 7, 3, 0, 0, 0, 0, 4, 0, 4], [-5, 0, 0, 0, 0, 1, 4, 0, -6, 0, 10, 9, 1, -2, -4, 0], [-2, -2, -2, -4, 0, 1, -3, 0, 0, 2, 0, 1, -1, 0, -3, 2], [9, 0, 18, -10, -10, -9, 18, -3, -5, 5, -5, 0, 0, 2, 0, 2], [2, 0, 0, -4, 0, 0, 0, -14, 0, -4, 0, 2, 0, 0, 0, 0], [-1, -1, -1, 5, 0, 2, -6, -5, 0, -10, 0, 5, -2, 0, -6, -5], [0, -13, 0, 10, -10, 0, 0, -4, -4, -10, -10, 11, -13, 8, -22, -8], [6, -6, 6, 0, -2, 0, 6, 0, -4, 0, 0, 0, 0, -2, -6, 0], [0, -8, 0, 8, -8, 0, 0, 1, -4, -4, -4, 4, -4, 3, -18, -3], [-9, 1, 1, 4, 0, 0, -4, -10, 0, 0, 0, -5, 0, 0, -4, -7], [2, 0, 0, 0, -8, 0, 2, 0, -4, 0, -4, 0, 0, 8, -2, 0], [0, -1, 0, -2, 2, 0, 0, 7, -7, -2, -2, 0, 1, 0, 0, 0], [3, 0, 0, 0, 0, -5, 2, 0, 6, 0, 0, -1, -5, -3, -2, 0], [0, 2, 0, 3, -3, 0, 0, -3, -1, -6, -6, -2, 4, 4, 4, -4], [4, 0, 0, 0, 0, -7, 3, -10, 0, 6, 0, 1, 7, 0, 3, 16], [3, 0, 0, 0, 0, 0, 3, 12, 0, 0, 0, 0, 0, 12, 0, 12], [0, 13, 0, -6, 6, 0, 0, 0, 3, 0, 0, -12, 0, -3, 1, 3], [0, 7, 0, 6, -6, 0, 0, 3, 3, -6, -6, -12, 7, -6, 24, 6], [-3, 0, 11, -4, -4, 11, 0, -4, -4, -4, 4, 0, 0, 0, 0, 0], [4, 0, 4, -4, -4, -8, 12, 4, -1, 8, -8, 0, 0, 5, 0, 5], [-6, 0, 0, 0, 2, 0, -12, 0, 2, 0, 2, -6, 0, -4, 12, 0], [0, 0, 0, 0, 0, 0, 0, 13, -17, 9, 9, -10, 10, 4, 10, -4], [-8, 4, -4, 0, 2, 2, -10, 0, 1, 0, 1, -2, 2, -5, 10, 0], [0, 6, 0, 0, 0, 0, 0, 8, -8, 0, 0, 5, -6, 0, 0, 0], [-10, 0, 0, -4, 0, 0, -10, 14, 0, 2, 0, 0, 0, 0, -10, 16], [2, 0, 10, -1, -1, -10, 4, -4, 4, 1, -1, 0, 0, -8, 0, -8], [4, -3, 3, 0, -4, 3, 0, 0, 1, 0, 4, -4, 3, 0, 0, 0], [1, 0, 0, 0, 0, 5, -6, -8, 0, -11, 0, 7, -5, 0, -6, -3], [0, 6, 0, -3, 3, 0, 0, 7, -2, 0, 0, -10, 0, -5, -4, 5], [4, 4, 4, -6, 0, -8, 2, 6, 0, 12, 0, 2, 8, 0, 2, 2], [10, 0, 6, -3, -3, 6, 0, -7, -7, -3, 3, 0, 0, 0, 0, 0], [0, 0, 0, -6, 6, 0, 0, -12, 3, 3, 3, 0, 0, 9, -4, -9], [9, 2, -2, 0, -1, 2, 18, 0, 1, 0, -1, 9, 2, -2, -18, 0], [0, 0, 0, 0, 0, 0, 0, 5, -5, 5, 5, 7, 1, 0, -3, 0], [-27, 0, -1, 4, 4, 0, -14, 2, 3, 0, 0, 0, 0, -1, 0, -1], [8, 0, 0, 0, 0, -8, 0, 0, -14, 0, -14, -8, -8, 14, 0, 0], [8, 0, -10, 10, 10, 5, 3, -19, 5, -5, 5, 0, 0, -24, 0, -24], [-12, 2, 2, -1, 0, 2, 0, -21, 0, -1, 0, -12, -2, 0, 0, 0], [-2, -2, -2, -3, 0, 4, -14, 3, 0, 6, 0, 12, -4, 0, -14, -6], [3, -1, 1, 0, -5, 0, 2, 0, -12, 0, 0, -1, 0, -7, -2, 0], [0, 6, 0, 4, -4, 0, 0, 3, -2, -2, -2, -3, 3, -1, 24, 1], [0, 0, 0, -2, -2, 0, 6, 2, -4, 4, -4, 0, 0, 6, 0, 6], [9, 1, 1, 0, 0, 0, 5, 12, 0, 0, 0, 4, 0, 0, 5, 6], [-10, -1, -1, 4, 0, 1, -20, -8, 0, -4, 0, 10, -1, 0, -20, -16], [-5, 0, 0, 0, 0, 9, -4, 0, 20, 0, 4, 1, 9, -12, 4, 0], [0, -2, 0, -3, 3, 0, 0, 3, 1, 6, 6, -12, -4, -4, 10, 4], [-6, 0, -4, -12, -12, 4, -12, 2, -2, 12, -12, 0, 0, 4, 0, 4], [-3, 0, 0, 0, 0, -9, 12, 1, 0, 4, 0, -15, 9, 0, 12, 3], [-4, 4, -4, 0, 3, 8, -8, 0, -19, 0, 6, -4, 8, 16, 8, 0], [12, -4, -4, -2, 0, -4, 0, 8, 0, -2, 0, 12, 4, 0, 0, 0], [0, -19, 0, 11, -11, 0, 0, -5, -3, 0, 0, 15, 0, 8, -4, -8], [4, 0, -8, 2, 2, -8, 0, -9, -9, 2, -2, 0, 0, 0, 0, 0], [-4, 0, -4, 4, 4, 8, -1, -4, 2, -8, 8, 0, 0, -6, 0, -6], [2, -6, 6, 0, 0, -6, 4, 0, -2, 0, 0, 2, -6, 4, -4, 0], [1, -5, -5, -2, 0, 0, -2, -32, 0, 0, 0, 3, 0, 0, -2, -15], [-6, -8, 8, 0, -2, -4, -2, 0, -1, 0, -1, 4, -4, 0, 2, 0], [0, -4, 0, -2, 2, 0, 0, 6, -6, -2, -2, 14, 4, 0, 0, 0], [-9, 0, -5, 3, 3, 0, -7, -17, -7, 0, 0, 0, 0, -10, 0, -10], [-6, -3, 3, 0, -4, 3, 0, 0, 14, 0, 4, 6, 3, 0, 0, 0], [8, 0, 0, 0, 0, 2, -10, 1, 0, -7, 0, 18, -2, 0, -10, -8], [1, -1, 1, 0, -3, -2, 4, 0, 20, 0, -6, 3, -2, -17, -4, 0], [-14, 0, 0, 0, 0, 18, -4, 2, 15, -11, 11, 0, 0, -13, 0, -13], [-7, 0, -7, 5, 5, -7, 0, 8, 8, 5, -5, 0, 0, 0, 0, 0], [0, 14, 0, 4, -4, 0, 0, -12, -2, -2, -2, -7, 7, 14, 3, -14], [0, 0, 0, 0, 0, 0, 0, 3, -8, -2, -2, -4, -12, 5, -4, -5], [4, 15, 15, -8, 0, -15, 8, 1, 0, 8, 0, -4, 15, 0, 8, 2], [0, 0, 0, 0, 0, 10, -10, 0, -13, 0, 9, -10, 10, 2, 10, 0], [0, -5, 0, 1, -1, 0, 0, -1, 5, -2, -2, 11, -10, -4, -16, 4], [26, 1, -1, 0, 2, -1, 0, 0, -9, 0, -2, -26, -1, 0, 0, 0], [0, 0, -12, 2, 2, 6, -6, -23, 1, -1, 1, 0, 0, -24, 0, -24], [21, -1, -1, 2, 0, -1, 0, -10, 0, 2, 0, 21, 1, 0, 0, 0], [-1, 0, 0, 0, 0, -1, 2, -7, -6, -8, 8, 0, 0, -1, 0, -1], [0, 21, 0, 0, 0, 0, 0, -1, -1, 0, 0, -8, 21, 2, 16, -2], [-8, 12, -12, 0, 5, 0, -10, 0, -3, 0, 0, -2, 0, -8, 10, 0], [-5, -7, -7, 11, 0, 0, -6, 5, 0, 0, 0, 1, 0, 0, -6, -3], [12, 12, 12, -16, 0, -12, 24, 8, 0, 16, 0, -12, 12, 0, 24, 16], [0, -2, 0, 2, -2, 0, 0, -2, -14, -4, -4, -11, -4, 16, 9, -16], [22, 0, 9, 1, 1, -9, 44, -2, 2, -1, 1, 0, 0, -4, 0, -4], [-18, 0, -8, 8, 8, 4, -22, 4, 4, -4, 4, 0, 0, 0, 0, 0], [-2, 2, -2, 0, -1, 4, 7, 0, -18, 0, -2, 9, 4, 19, -7, 0], [-8, 5, 5, -5, 0, 5, 0, 5, 0, -5, 0, -8, -5, 0, 0, 0], [0, 6, 0, -2, 2, 0, 0, -10, 6, 0, 0, -10, 0, 4, -4, -4], [0, -14, 0, 11, -11, 0, 0, -11, -11, -11, -11, -10, -14, 22, 20, -22], [-9, 3, -3, 0, 5, 3, -18, 0, 0, 0, 5, -9, 3, 0, 18, 0], [8, -2, -2, 4, 0, 0, 3, 28, 0, 0, 0, 5, 0, 0, 3, 12], [0, 0, 0, 0, 0, 0, 0, -4, -2, -10, -10, 24, -8, 6, -16, -6], [-9, -6, 6, 0, -8, -3, -6, 0, -4, 0, -4, 3, -3, -11, 6, 0], [2, 2, 2, -10, 0, -1, 3, 27, 0, 5, 0, -1, 1, 0, 3, 32], [7, 0, 10, -8, -8, -10, 14, 14, -14, 8, -8, 0, 0, 28, 0, 28], [-4, -4, 4, 0, -12, 4, 0, 0, -20, 0, 12, 4, 4, 0, 0, 0], [1, 0, 0, 0, 0, -6, 5, 14, 0, 23, 0, -4, 6, 0, 5, 9], [-2, -2, -2, -3, 0, 4, -4, 3, 0, 6, 0, 2, -4, 0, -4, -11], [10, 0, -6, -3, -3, -6, 0, -5, -5, -3, 3, 0, 0, 0, 0, 0], [0, 8, 0, -2, 2, 0, 0, -3, 1, 1, 1, -4, 4, 2, 28, -2], [0, 0, 0, 0, 0, 0, 0, -8, 22, 6, 6, 12, 6, -14, -3, 14], [-2, -16, -16, 14, 0, 16, -4, -12, 0, -14, 0, 2, -16, 0, -4, -24], [8, 0, 8, 4, 4, 0, 8, 12, 8, 0, 0, 0, 0, 4, 0, 4], [4, 16, 16, -2, 0, -8, 12, -22, 0, 1, 0, -8, 8, 0, 12, -21], [4, 8, -8, 0, 0, -8, 0, 0, 4, 0, 0, -4, -8, 0, 0, 0], [-11, -6, -6, 11, 0, -6, 0, 25, 0, 11, 0, -11, 6, 0, 0, 0], [11, 0, 0, 0, 0, -3, -8, 3, -6, 12, -12, 0, 0, 9, 0, 9], [7, 7, 7, 2, 0, -14, 13, -2, 0, -4, 0, -6, 14, 0, 13, -14], [-19, -15, 15, 0, -14, 0, -2, 0, -3, 0, 0, 17, 0, 11, 2, 0], [6, 0, 6, -9, -9, -12, -16, 9, -8, 18, -18, 0, 0, 17, 0, 17], [-22, -2, -2, -2, 0, 0, -12, 2, 0, 0, 0, -10, 0, 0, -12, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_490_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_490_2_l_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_490_2_l_c(:prec:=16)
    chi := MakeCharacter_490_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_490_2_l_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_490_2_l_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_490_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, -96, 288, -576, -276, 1968, 3528, 3780, 2357, 564, 0, -84, -45, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;